On the geometry, preemptions and complexity of multiprocessor and shop scheduling

In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called preemption and machine-dependency graphs for preemptive multiprocessor and shop scheduling problems, respectively. In a natural manner, we call a scheduling problem acyclic if the corresponding graph is acyclic. There is a substantial interrelation between the structure of these graphs and the complexity of the problems. Acyclic scheduling problems are quite restrictive; at the same time, many of them still remain NP-hard. We believe that an exhaustive study of acyclic scheduling problems can lead to a better understanding and give a better insight of general scheduling problems.